On the monodromy group of the family of smooth plane curves

2016/10/16 by Salter, Nick
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.04920

Abstract

We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an invariant known as a "n-spin structure" constrains the image in ways not predicted by previous work of Beauville. Second, we show that for degree d=5, our invariant is the only obstruction for a mapping class to be contained in the image. This requires combining the algebro-geometric work of Lönne with Johnson's theory of the Torelli subgroup of the mapping class group.

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